How to tell if something has a common factor

  • Thread starter rsala004
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In summary, if we have two numbers, 6 and 9, and we break them down into sets of prime factors {2,3} and {3}, then their intersection is empty, meaning they have no common factors. This also applies to any two numbers, P and Q, if their sets of prime factors have an empty intersection, then P/Q is an irreducible fraction. This is also true if the sets of divisors have {1} as their intersection. To understand the theory behind this, one can look into Euclid's Algorithm and the Chinese Remainder Theorem.
  • #1
rsala004
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if we have 6 and 9

and we break them down to sets of prime factors {2,3} , and {3}

if the intersection of the 2 sets is empty..does this mean that numbers have no common factors?

or in more specific to my interest...

if we have P and Q and we have their sets of prime factors, if the intersection is empty does this mean P/Q is an irreducible fraction?

or is this only true for their sets of non-zero prime+composite factors?

thanks
 
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  • #2
rsala004 said:
if we have 6 and 9

and we break them down to sets of prime factors {2,3} , and {3}

if the intersection of the 2 sets is empty..does this mean that numbers have no common factors?

or in more specific to my interest...

if we have P and Q and we have their sets of prime factors, if the intersection is empty does this mean P/Q is an irreducible fraction?

or is this only true for their sets of non-zero prime+composite factors?

thanks
Your main question has yes as an answer.

I don't know what you mean by
this only true for their sets of non-zero prime+composite factors
 
  • #3
rsala004 said:
if we have P and Q and we have their sets of prime factors, if the intersection is empty does this mean P/Q is an irreducible fraction?

or is this only true for their sets of non-zero prime+composite factors?

thanks


By definition, a set of prime factors won't contain any composites.
 
  • #4
I'm going to guess the question and then answer what I thought the question was.

Question: Given positive integers m and n, if the set M of primes dividing m and the set N of primes dividing n are disjoint (have an empty intersection), is m/n an irreducible fraction? [If this is false, is it at least the case that if the set M' of divisors of m and the set N' of divisors of n have {1} as their intersection, is m/n an irreducible fraction?]

Answer: Yes, if there are no primes in common than m/n is irreducible.
 

1. How do I know if two numbers have a common factor?

Two numbers have a common factor if they have at least one number that can evenly divide both of them. This means that the remainder of the division should be 0 for both numbers. For example, the numbers 12 and 18 have a common factor of 6 since both 12÷6=2 and 18÷6=3 have a remainder of 0.

2. What is the largest common factor of two numbers?

The largest common factor of two numbers is also known as the greatest common factor (GCF). It is the largest number that can divide both numbers without leaving a remainder. It is often found by listing out all the factors of the two numbers and finding the largest one that they have in common. For example, the GCF of 24 and 36 is 12 since the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24 and the factors of 36 are 1, 2, 3, 4, 6, 9, 12, and 36, and the largest number they have in common is 12.

3. How can I quickly tell if two numbers have a common factor?

One quick method is to find the prime factorizations of the two numbers. If they have at least one prime factor in common, then they have a common factor. For example, the prime factorization of 24 is 2 x 2 x 2 x 3 and the prime factorization of 36 is 2 x 2 x 3 x 3. They both have 2 and 3 as prime factors, so they have a common factor of 12.

4. Can fractions have common factors?

Yes, fractions can have common factors. To find the common factors of two fractions, you can simplify them to their lowest terms and then find the GCF of the numerators and denominators. For example, the fractions 8/12 and 12/18 both simplify to 2/3, and the GCF of 8 and 12 is 4, so their common factor is 4/6.

5. How can I use common factors to simplify fractions?

To simplify fractions using common factors, you can divide both the numerator and denominator by their GCF. This will result in an equivalent fraction that is in its simplest form. For example, the fraction 20/30 has a GCF of 10, so dividing both the numerator and denominator by 10 results in the simplified fraction of 2/3.

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